When I went to Italy, I can’t recall ever being offered tap water. It was always fancy bottled water that was served at restaurants. Granted, this was many years ago, so I might be remembering it incorrectly. Is the tap water safe to drink? Is there anywhere that it wouldn’t be safe?
I think it’s more of a customary thing than a health risk. Water is just always served in a bottle in restaurants. There are water fountains you can use when you’re out sightseeing. I’m guessing that more rural areas wouldn’t be as safe as tourist towns, but Italy is a modern country with potable water unless a sign states otherwise.